Exploring toroidal anvil profiles for larger sample volumes above 4 Mbar

With the advent of toroidal and double-stage diamond anvil cells (DACs), pressures between 4 and 10 Mbar can be achieved under static compression, however, the ability to explore diverse sample assemblies is limited on these micron-scale anvils. Adapting the toroidal DAC to support larger sample volumes offers expanded capabilities in physics, chemistry, and planetary science: including, characterizing materials in soft pressure media to multi-megabar pressures, synthesizing novel phases, and probing planetary assemblages at the interior pressures and temperatures of super-Earths and sub-Neptunes. Here we have continued the exploration of larger toroidal DAC profiles by iteratively testing various torus and shoulder depths with central culet diameters in the 30–50 µm range. We present a 30 µm culet profile that reached a maximum pressure of 414(1) GPa based on a Pt scale. The 300 K equations of state fit to our P–V data collected on gold and rhenium are compatible with extrapolated hydrostatic equations of state within 1% up to 4 Mbar. This work validates the performance of these large-culet toroidal anvils to > 4 Mbar and provides a promising foundation to develop toroidal DACs for diverse sample loading and laser heating.

These previous toroidal and double-stage anvil studies have primarily focused on reaching the maximum possible pressure in the DAC and attempts to heat the sample or load the sample embedded in thermal insulation are limited 13,14,19 .Probing the interiors of super-Earths and sub-Neptunes not only requires the generation of 3-6 Mbar pressures, but it also requires synthesis of materials at high temperatures in complex petrology regimes.
Here we present an expanded study of toroidal anvil profiles that accommodate large enough sample volumes for multi-layer sample packages and synchrotron or laboratory laser heating specifications.The toroidal anvil designs tested here have 30-50 µm culet diameters.After modifying the culet shape, torus depth, and shoulder depth to optimize the attainable pressures with these anvils, we present a profile that reached a maximum pressure of 414 (1) GPa based on the Pt scale 24 .We validate the performance of these anvils by measuring the volumes of gold and rhenium, fitting an equation of state, and comparing the fits to previous studies.Going forward, further development of the anvils, sample configuration, and heating methods may be tested for routine high temperature experiments, as the 12-18 µm diameter sample chambers tested with these anvils offer new possibilities for experiments with diverse samples at extreme pressures and temperatures.

Toroidal anvil design optimization
The starting anvils for each test were standard cut with 200 µm diameter culets.Toroidal shapes were milled with the focused ion beam incrementally modifying the culet diameter, torus depth, shoulder depth, and bevel of the central culet.As the behavior of toroidal anvils with varying specifications for culets larger than 25 µm has not been well characterized, we initially scaled the culet and torus-depth proportions linearly from (19) and (20) to a 30-50 µm culet.With this, we tested toroidal shapes with torus depths in the range of 9-20 µm (Fig. 1) and with shoulder depths ranging from 2/3 to 3/4 the torus depth 20 .Specific details for each anvil tested are provided in Table 1.The various proportions tested are divided into three shapes: Shape 1 contains a 50 µm culet with a shoulder depth that is 2/3 the torus depth, Shape 2 consists of a 50 µm culet with a shoulder depth that is 3/4 the torus depth, and Shape 3 consists of a 30 µm culet that bevels to 50 µm and a shoulder depth that is 3/4 the torus depth.A schematic of these three tested anvil shapes is provided in Figure S1 and the anvil shape that performed to the highest pressures in this study is shown in Figure 1a.
Figure 1b provides a scanning electron microscopy image of a Shape 3 toroidal anvil with the torus-depth measurement labeled.Each anvil was milled until the torus depth measured in cross section reached the desired depth given in Table 1 and shown in Fig. 1b.The torus depths were measured in the SEM using a trigonometric relation while the sample stage was tilted at a 52° angle for milling.In each case, the torus depths reported here can be reproduced by measuring the torus-to-culet-bevel distance in cross section intermittently during milling following the example in Fig. 1b.When repeatedly making this measurement on the same anvil, torus depths were measured within an uncertainty of ~ 1 µm.By tracking the milling time per anvil, consistent anvil profiles can then be reproduced setting the milling time with the same milling parameters.As depth measurements in the SEM are challenging due to factors such as choosing the start and end points, correcting for the perspective view, and changes in the anvil surface due to redeposition, we additionally evaluated the dimensions of our tDACs in profile view optically after milling (Fig. S2).For a Shape 3 anvil with a torus depth of ~ 20 µm measured in cross section in the SEM, we measured a ~ 14 µm shoulder depth optically with a microscope with 0.9 µm resolution at the highest magnification (Fig. S2).For a ~ 20 µm torus depth, a Shape 3 anvil should have a 15 µm shoulder depth (Table 1).The uncertainties in depth measurement observed here are small relative to the range of torus depths tested (9-20 µm) and are within the range of anvil torus depths that perform satisfactorily (18-20 µm) as discussed below.To produce the anvil shown in Figure 1b and all anvils shown in Table 1, we also scaled the bitmap to 250 × 250 µm centered on a 200 µm flat anvil prior to milling.The region of the bitmap that overlaps with the anvil pavilion therefore shows a steep drop-off in the milled anvil, providing an additional bevel to the shoulder to counteract anvil deformation under high loading (Fig. 1a,b).The edge of the FIB milled section exhibits a sharp artifact at the intersection of the bitmap slope with the anvil pavilion.By extending the bitmap out onto the pavilion, this sharp edge is also offset significantly lower than the culet such that it does not cause anvil failure, as confirmed by our absorption scans collected under compression (Fig. 1c, Fig. S3). Figure 1a illustrates the starting 200 µm and the idealized bitmap profile, while the resulting profile that includes the artifacts of milling on the pavilion and milling at a high 2.5 µA current in the FIB in general is shown in Fig. 1b.We point out that Shapes 2 and 3 have a stepped shoulder profile in the raw bitmap (Fig. 1a, Fig. S1), but given the scaling of the bitmap onto the pavilion in this region, it is likely that this feature is not effective here but may prove useful if the bitmaps are scaled concentrically with the culet only (Fig. 1a, Fig. S1).
For a 50 µm culet, Shape 1 and 2 profiles with 9-15 um torus depth failed under 2 Mbar due to the shoulders of the anvils deforming under load and cutting through the gasket (Fig. S3a,b).Once the torus depth was increased to 18 µm for a 50 µm culet, the shoulders of the anvils still deformed under compression but were no longer the cause of anvil failure.Instead, the cupping of the culets caused anvil failure still under 2 Mbar (Fig. S3c).A bevel was then introduced to the culets in Shape 3 (Fig. 1a), whereby the 50 µm culet was beveled down to a 30 µm central culet.For this shape, torus depths of 16, 18, and 20 µm were tested (Table 1, Fig. 1).The 16 µm torus depth tested on F001 survived only to ~ 200 GPa, but the 18 µm torus depth and 20 µm torus depth anvils compressed between 249 and 414 GPa (Table 1).It should be noted that C172 was compressed to 249 GPa and held there to test laser heating (Supplemental text S3), and therefore, this pressure is an underestimation of the performance of the anvils in this experiment.The Shape 3 anvils with 20 µm torus depths achieved the highest pressures in this work with half of the tested anvils achieving between 3-4 Mbar (Table 1).A more rigorous evaluation of the pressures obtained and their uncertainties based on the indexed volumes of Pt, Au, and Re are provided for samples C172 and C173 and discussed in the following sections.For C173, the shoulders of the anvils exhibited moderate deformation below 4 Mbar (Fig. 1c), but this deformation did not pinch out the gasket Table 1.Toroidal anvil specifications and performance evaluated for each sample.The torus depth and shoulder depth are approximated from the measurement of the torus depth -culet-bevel in cross-section in the SEM.

Shape Sample Culet (c)/bevel (b) (μm) Torus depth (t) (μm) Shoulder depth (s) (μm) Loading
Maximum pressure (GPa) www.nature.com/scientificreports/ in this compression cycle, and the anvils supported pressures > 4 Mbar.It is likely that the deformation of the culets caused anvil failure at peak pressures (Fig. 1c), as Shape 3 anvils consistently fail via a vertical crack through the culet.We recognize that the performance of these anvils also depends on other factors such as alignment of the anvils and starting sample density, however, these initial tests indicate that the Shape 3 anvils do repeatedly perform to the highest pressures compared to the other tested shapes.A bitmap used to make the Shape 3 anvils is provided (Supplementary Data S1).At central pressures equal to ~ 3.7 Mbar, determined from Pt 24 , we collected horizontal and vertical X-ray scans of sample C173 to assess the pressure distribution across the anvils.Tracking the Re pressure 25 across the gasket and sample, pressures increased from ~ 20 to 60 GPa in the shoulder and torus region of the anvils before spiking up to 3.7 Mbar on the 30 µm culets (Fig. S3).The pressure gradient across the culets does not exceed ~ 20 GPa at 3.6 Mbar (Fig. S4).The observed confining pressure, ~ 40-60 GPa, is comparable to that reported for the double-stage diamond anvils [15][16][17][18]21 , but less than half that reported for other toroidal diamond anvils 20 .

Volume and pressure determination
The pressure exerted by the toroidal anvils for a nonhydrostatic loading was monitored and validated by measuring the volumes of platinum and gold using synchrotron X-ray diffraction for samples C172 and C173 (Fig. 2).Rhenium was additionally observed from the gasket in sample C173 and its volume was measured (Fig. 2).Volumes were calculated for fcc Pt and Au based on the (111) lattice plane (Fig. 3, Dataset S2).Previous hydrostatic 26 and non-hydrostatic 19,23 DAC compression studies have established that the (111) lattice plane of fcc metals including gold and silver is least affected by uniaxial stress while the (200) lattice plane is most affected by uniaxial stress.Therefore, the volume for gold and platinum calculated from the (111) lattice plane is closer to a hydrostatic approximation compared to the volume determined from multiple lattice planes.Additionally, as shown in Fig. 3, above 275 GPa, the fcc Pt and Au peaks overlap to the point where the maxima of Au cannot be differentiated from the higher intensity Pt peak.Once peak overlap initiated, we then held the full-width-halfmaximum of the Pt and Au peaks fixed to obtain a consistent peak fitting with further pressurization to 414 GPa.Rhenium volumes were determined from the measured (100), (101), and (102) d-spacings.
We chose Pt as the pressure calibrant, as the volumes indexed from the (111) and ( 200) lattice planes of Pt do not show any significant deviation with pressure and agree within < 0.7% across this study (Fig. S5, Supplementary dataset S2).As such, the uniaxial stress calculated for Pt based on these volume differences 27 do not show a trend with pressure and reach at most 1.6 GPa across this study (Fig. S5, Supplementary text S2).These results suggest that Pt compresses relatively isotropically and does not maintain significant uniaxial stress, despite the nonhydrostatic compression environment and is a reliable ultra-high-pressure standard.For pressure calibration, the volume obtained from the (111) peak of platinum was considered, as it is the closest approximation to the hydrostatic volume based studies of analogous fcc metals 19,23,26 .We chose to calibrate our platinum volumes to the recent ramp compression Pt scale 24 , as this scale is calibrated beyond the pressures of this study and is compatible with previous DAC studies [28][29][30] (Fig. S6).For samples C172 and C173, the application of the Pt pressure scale 24 results in a maximum obtained pressure of 249.2(3)GPa and 414(1) GPa, respectively.

Equation of state of gold to 4.1 Mbar
Figure 3a shows the evolution of gold volumes indexed from the (111) lattice plane with platinum pressure compared to the gold volumes indexed from the (111) lattice plane reported by (19) (nonhydrostatic) and ( 28) (helium pressure-transmitting medium).The gold volumes indexed for samples C172 and C173 show a smoothly decreasing trend with pressure and are in agreement with each other and with previous hydrostatic compression studies 26 .Our indexed volumes are marginally larger than that reported by (19), which may reflect that our compression conditions were less hydrostatic or it may reflect differences in the pressure calibrants used.
We then fit a Vinet equation of state to our Au (111) volumes following the form: We provide several fittings in Table 2. Considering the merging of the Au and Pt peaks with pressure, where the Au (111) peak maxima cannot be independently observed above 275 GPa (Fig. 2), we fit equations of state to our data up to 275 GPa as well as up to 414 GPa for comparison.In all scenarios, we fixed the ambient volume to 16.962 Å 3 from 31 .We report fits with the ambient isothermal bulk modulus (K 0 ) fixed to the ultra-sonically derived value of 167 GPa 32 and as a free parameter (Table 2).Despite the peak overlap observed in Au and Pt at the conditions reached in this study, comparison of the EoS fits with V 0 = 16.962Å 3 and K 0 = 167 GPa up to 275 GPa and 414 GPa reveals that the fitted K' equals 6.05(1) and 6.075 (6), respectively (Table 2).These results support that even adopting a conservative equation of state fit to the non-overlapping data, our fitted volumes in the 275-414 GPa range are still reasonably described by these fitting parameters.Figure 4 includes the equation of state fit to our gold V 111 volumes up to 414 GPa with V 0 = 16.962Å 3 , K 0 = 167 GPa, and K' 0 = 6.075 (6).Pressure residuals to this fit reach at most a 4% deviation from the reported Pt pressure (Fig. 4).The K 0 ' value for this nonhydrostatic fit is 3% larger than K 0 ' value reported in the hydrostatic compression study (26).Equation of state fits for Au from previous studies 19,24,26,30,[33][34][35] are also included in Table 2 for comparison.

Equation of state of rhenium to 4.1 Mbar
In Experiment C173, the volumes of rhenium under nonhydrostatic loading conditions from the gasket were also determined up to 414(1) GPa from the measured (100), (101), and (102) lattice spacings.Rhenium volumes show a smoothly decreasing trend with pressure and are calibrated to the Pt pressure scale 24 (Fig. 4).Comparisons to previous P-V relationships of rhenium up to multi-megabar pressures reveal that the compression data (1) determined in this study are compatible with the compression curve of rhenium in helium reported by 25 , and not compatible with the P-V relationship reported via double stage DAC compression experiments in He and Ne by 13 (Fig. 4).A Vinet equation of state was fit to the data assuming an ambient volume of 14.733 Å 3 per atom after 25 and fixing the ambient bulk modulus to the ultra-sonic value of 360 GPa 36 .As all tDAC samples started at some non-zero pressure, we assumed an ambient volume following 25 .This fit is shown in Fig. 4, and fits reported by additional studies 13,25,[36][37][38][39][40][41] are provided for further comparison in Table 3.We conducted an EoS fit to our Re data allowing both K 0 and K′ as fitting parameters as well and listed these results in Table 3.  102) lattice spacings calibrated to the platinum standard 24 for this study (orange squares).The Vinet equation of state fit to our data with K 0 fixed to 360 GPa 33 and K' 0 fit to 4.52( 3) is provided by the solid orange line.Pressure residuals to this fit are included in the bottom panel.For comparison, rhenium volumes reported by 25 , calibrated to ruby 42 , tungsten 42 , and helium 43 pressure standards are included (black open circles), and rhenium volumes reported by 13 calibrated to gold 33 are also included (grey open triangles).www.nature.com/scientificreports/

Equation of state comparisons for gold and platinum
The gold and rhenium data collected in this study provide an independent check and evaluation of the Au and Re pressure scales to > 4 Mbar.Our results for the compression of gold under nonhydrostatic conditions in the toroidal diamond anvil cell agree with a multitude of previous static and dynamic equations of state of gold within ~ 1% by volume up to 4 Mbar (Fig. 6a) 19,24,26,30,33,34 .This volume difference would result in a pressure difference of ~ 20 GPa at 4 Mbar (~5 %) (Fig. 6b).
The equation of state fit to the rhenium volumes determined in this study is within ~ 0.75% volume and ~ 20 GPa of the extrapolated equation of state of rhenium in helium 35 to 4 Mbar (Fig. 6c,d).Compared to other Re equation of state studies 13,[37][38][39][40][41] , our results are most similar with the first principles EoS by 41 .Alternatively, the equations of state determined experimentally by 13 and theoretically by 39 reports ~ 100 GPa higher pressures than Figure 5. Cell pressure with membrane pressure (bar) or screw rotation (deg) for the highest-pressure cell run in this study (C173) compared to other toroidal anvil and standard bevel anvil studies 19,20,45 .The colors of the lines and data are associated with the relevant x-axes, depending on whether the study reported the loading curve from membrane compression or manual screw turning.The shapes of the anvils associated with each pressure-loading curve are provided in the inset..(a) compared to the equation of state fit to gold in helium 26 .(c) Percent volume and (d) pressure residuals of the rhenium equation of state fits from this study 13 , and [37][38][39][40][41] compared to the equation of state fit to rhenium in helium 25 .The solid lines represent the data coverage of each study, and the dashed lines represent the extrapolations of the fits.www.nature.com/scientificreports/ the hydrostatic extrapolation 25 by 320 GPa (Fig. 6).The equations of state reported by 37,40 diverge significantly in the negative direction above ~ 130 GPa from the hydrostatic extrapolation 25 .The results from this study indicate that under nonhydrostatic loading conditions up to > 4 Mbar, rhenium in our experiments behaves more similarly to 25 compared to these previous experimental and computational reports.Results from this study help to validate rhenium under nonhydrostatic compression as a pressure calibrant by providing data in support of recent works using this pressure scale 25 to 6 Mbar 19 .Unifying these P-V relationships to a single pressure scale will further improve comparisons of the gold and rhenium equations of state discussed here, but agreement between our data and extrapolations of equations of state of gold and rhenium in helium 25,26 , validate the performance of the large-volume toroidal anvils tested here to > 4 Mbar.With further tests of laser heating using such large-volume toroidal anvils, we are close to generating P-V-T data of key pressure standards over a wider pressure-temperature range that would lead to an internally consistent pressure scale for planetary materials applications.

Conclusion
In this study, we aimed to present a toroidal anvil profile that optimizes sample volume for compression studies above 300 GPa.Increasing sample size aids in reaching ultrahigh pressure-temperature conditions where many exotic physics and chemistry phenomena have been predicted and that are relevant to Earth and exoplanetary interiors.We explored the relationship between culet diameter, torus depth, and shoulder depth for anvils with central culets in the 30-50 µm range.Using toroidal anvils with 30 µm culets, torus depths of 20 µm and shoulder depths of 15 µm, we achieved a maximum pressure of 414(1) GPa based on the platinum pressure scale.Equations of state fitted to our gold (111) and rhenium volumes agree within 1% and 0.75% by volume of previously reported hydrostatic equations of state for these materials extrapolated to 4 Mbar, respectively.These results cross-validate that the toroidal anvils tested in this study reached > 4 Mbar and provide a promising basis to further develop these large-volume toroidal anvils for exploring exotic materials and planetary petrology at high temperatures in the 3-4 Mbar range.

Anvil fabrication
Toroidal diamond anvils were fabricated at the Carnegie Institution for Science using an FEI Helios plasma focused ion beam (PFIB) G4 with a xenon source.Type Ia, standard-cut 200 µm culet diamond anvils sourced from Almax easyLab were used as the base for toroidal milling.The toroidal profiles were imported as bitmaps into the PFIB and scaled to a 250 µm by 250 µm milling region centered on the culet.Anvil milling was performed using an ion energy of 30 kV and beam current of 2.5 µA.As the FIB milling depth for this instrument is calibrated to silicon, the milling process for each anvil was monitored manually by intermittently pausing the milling and measuring the inner culet height using the "cross-section" setting.The anvils were considered complete when the desired torus depth was achieved within ~ 1 µm.The torus depth was repeatedly measured as shown in Fig. 1b, from the base of the torus where redeposition on the walls of the culet is observed to the edge of the culet bevel where the redeposition from anvil milling ends.Additionally, all anvils were milled radially outwards to reduce redeposition on the anvil tip.Total milling time typically ranged from 1 to 1.5 h per diamond.The bitmap used for the highest-pressure experiments reported here is included in the supplementary material (Supplementary file S1).

Sample preparation
The milled toroidal diamonds were mounted on tungsten carbide seats.Sample chambers were prepared by indenting rhenium or tungsten gaskets to 12-15 µm thick, drilling an 8-12 µm sample chamber, and filling the sample chamber with pressed foils of platinum and gold as pressure markers for X-ray diffraction.Select samples were also loaded with SiO 2 to test multi-layer sample configurations for future laser heating as well as cBN inserts to increase sample thickness and reduce the diffraction signal from the gasket (Table 1).

Anvil performance evaluation using X-ray diffraction
The performance of the toroidal anvils was monitored using angle-dispersive synchrotron X-ray diffraction at Argonne National Laboratory, Sector 13 ID-D of the Advanced Photon Source.An X-ray beam measured at 2.5 µm × 3.54 µm full width, half maximum tuned to 42 keV was used to probe the sample chamber under compression.X-ray diffraction was collected on a CdTe 1 M Pilatus detector calibrated using a LaB 6 NIST powder standard.Upon compression, X-ray absorption scans of the milled region of the anvils were collected to monitor the deformation of the anvils with pressure.The pressure generated by the anvils was determined by collecting X-ray diffraction from platinum, gold, and rhenium on the central culet.X-ray exposure times of 1-2 s were used.Diffraction data was monitored during experiments using Dioptas 46 and processed using in-house OriginC codes.Pressure was calibrated using the equation of state of platinum 24 for measurements in the sample chamber and of rhenium 25 for measurements collected scanning across the gasket.

High temperature testing
On select samples, X-ray diffraction was collected during laser heating.Samples were heated with 100 W infrared fiber lasers in double sided configuration at Sector 13 ID-D, APS.The laser beam was shaped to a10 µm flat-top and aligned to the X-ray beam using the fluorescence of the sample 47 .The laser shutter was incrementally opened on the sample at a set laser power and exposure in the "burst" heating mode.The thermal emission collected from the central 6 µm of the laser flat top was recorded using an Action SP-2360 imaging spectrograph with a

Figure 1 .
Figure 1.(a) General schematic of the toroidal Shape 3 tested in this study on sample C173.The profile of the starting 200 µm standard anvil is included.The idealized bitmap profile, shaded in blue was scaled to 250 × 250 µm, such that the outer shoulder overlaps with the pavilion of the standard anvil.A sharp edge is produced at the boundary of the bitmap and is an artifact of the differing slopes between the bitmap and the diamond pavilion.(b) Example of the toroidal anvil Shape 3 milled with the focused ion beam showing the real profile that results from milling onto the pavilion and the sharp outer lip produced by the bitmap slope that is offset onto the pavilion.(c) Absorption scan of sample C173 with Shape 3 between 30 and 414 GPa measured at the center of the sample chamber based on the Pt scale 24 .

Figure 2 .
Figure 2. (a) X-ray diffraction pattern of sample C173 loaded with Pt and Au at maximum pressures of 414(1) GPa based on the platinum volume calibrated to the ramp compression scale 24 .(b) Integrated X-ray diffraction patterns of sample C173 loaded with Pt and Au showing diffraction from Pt, Au, and Re up to 414(1) GPa based on the platinum standard 24 .

Figure 3 .
Figure 3. (a) Evolution of gold volume calculated from the (111) plane for samples C172 (closed symbols) and C173 (open symbols) run in this study.A Vinet equation of state fit, with K 0 fixed to 167 GPa 32 and K' 0 fit to 6.075(6) is provided via the orange solid line.Pressure residuals to this EoS fit are included in the bottom panel.Gold volumes calculated from the (111) plane from (19) (green circles) and (26) (blue triangles) are also included for comparison.

Figure 4 .
Figure 4. Rhenium volumes calculated from the measured (100), (101), and (102) lattice spacings calibrated to the platinum standard24 for this study (orange squares).The Vinet equation of state fit to our data with K 0 fixed to 360 GPa33 and K' 0 fit to 4.52(3) is provided by the solid orange line.Pressure residuals to this fit are included in the bottom panel.For comparison, rhenium volumes reported by25 , calibrated to ruby42 , tungsten42 , and helium43 pressure standards are included (black open circles), and rhenium volumes reported by13 calibrated to gold33 are also included (grey open triangles).

Figure 6 .
Figure 6.(a) Percent volume and (b) pressure residuals of the equations of state fits from this study (V 111 )19,24,30,33,34 .(a) compared to the equation of state fit to gold in helium26 .(c) Percent volume and (d) pressure residuals of the rhenium equation of state fits from this study13 , and[37][38][39][40][41] compared to the equation of state fit to rhenium in helium25 .The solid lines represent the data coverage of each study, and the dashed lines represent the extrapolations of the fits. https://doi.org/10.1038/s41598-024-61861-2